Derived categories of genus one curves and torsors over abelian varieties
2024-04-04 10:30 - 11:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Jonathan Rosenberg, University of Maryland

Title: Derived categories of genus one curves and torsors over abelian varieties

Abstract: Studying orientifold string theories on elliptic curves or abelian
varieties motivates studying the derived category of coherent sheaves on
a genus one curve or a torsor over an abelian variety over the reals
(as opposed to the complex numbers).

In joint work with Nirnajan Ramachandran (to appear in MRL), we show that
a genus one curve over a perfect field determines a class in the relative
Brauer group of the Jacobian elliptic curve, and that there is a natural
Mukai-type derived equivalence between the original genus one curve
and the Jacobian twisted by the Brauer class. The proof extends to
torsors over abelian varieties (of any dimension).

Mirror symmetry for fibrations and degenerations of K3 surfaces
2024-04-11 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Alan Thompson (Loughborough University)
Title: Mirror symmetry for fibrations and degenerations of K3 surfaces
Abstract: In 2016, Doran, Harder, and I conjectured a mirror symmetric relationship between Tyurin degenerations and splittings of codimension 1 fibrations on Calabi-Yau manifolds. In this talk I will discuss recent work to make this conjecture rigorous in the case of K3 surfaces. I will give a precise definition of what it means for a Tyurin degeneration of K3’s to be mirror to a splitting of an elliptically fibred K3, and show that this definition enjoys the following compatibilities with existing mirror symmetric theories: 1) The general fibre of the Tyurin degeneration is mirror to the elliptically fibred K3, in the sense of Dolgachev-Nikulin. 2) Components of the Tyurin degeneration and pieces of the splitting satisfy a homological version of the (quasi-) Fano-LG correspondence. 3) Components of the Tyurin degeneration which are weak del Pezzo are mirror to pieces of the splitting that arise as restrictions of the corresponding lattice polarised LG models to discs. This is joint work with Luca Giovenzana.

Geometric local systems on very general curves
2024-04-18 10:15 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Aaron Landesman, MIT
Title: Geometric local systems on very general curves
Abstract: What is the smallest genus h of a non-isotrivial curve over the generic genus g curve? In joint work with Daniel Litt, we show h is more than $sqrt{g}$ by proving amore general result about variations of Hodge structure on sufficiently general curves. As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in the character variety parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.

The logarithmic double ramification locus
2024-04-25 10:30 - 11:30 | Virtual
Algebraic Geometry in String Theory Seminar
Speaker: Alessandro Chiodo, IMJ-Paris Rive Gauche (Jussieu)
Title: The logarithmic double ramification locus
Abstract: Given a family of smooth curves C -> S with a line bundle L on C, it is natural to study the locus of points x in S where L_x is trivial on C_x. When the family is stable, the definition can be extended, not directly on the base scheme S, but more naturally on a (logarithmic) blow-up S’ of S. The problem is in many ways analogue to the problem of defining a Néron model on the moduli space of stable curves (instead of a DVR). Over the past years, David Holmes and his collaborators pioneered a new approach on a logarithmic modification of the entire moduli space of curves. In this talk, we determine this logarithmic double ramification cycle and several variants and alternative presentations of it (work in collaboration with David Holmes).

Computing periods of hypersurfaces and elliptic surfaces via effective homology
2024-05-09 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Eric Pichon-Pharabod, Universite Paris-Saclay
Title: Computing periods of hypersurfaces and elliptic surfaces via effective homology
Abstract: The period matrix of a smooth complex projective variety X encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with sufficient precision of the entries of this matrix, called periods, allow to recover some algebraic invariants of the variety, such as the Néron-Severi group in the case of surfaces. In this talk, we will present a method relying on the computation of an effective description of the homology for obtaining such numerical approximations of the periods of hypersurfaces and elliptic surfaces.

Mirror symmetry and log del Pezzo surfaces
2024-05-16 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Franco Rota, University of Glasgow
Title: Mirror symmetry and log del Pezzo surfaces
Abstract: The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex geometry of a variety X (the B side) and the symplectic geometry of its mirror object Y (the A side). Motivated by this, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. I will discuss early results on the A side, using the language of pseudolattices to focus on the special case of a smooth degree 2 del Pezzo surface. This is joint work with Giulia Gugiatti.
